In honor of artist Mark Lombardi, we
define a Lombardi drawing to be a drawing of a graph in which the edges
are drawn as circular arcs and at each vertex they are equally spaced
around the vertex so as to achieve the best possible angular
resolution. We describe algorithms for constructing Lombardi drawings of
regular graphs, 2-degenerate graphs, graphs with rotational symmetry,
and several types of planar graphs. A program for the rotationally
symmetric case, the Lombardi
Spirograph, is available online.

We consider balloon drawings of trees, in which each subtree of the root
is drawn recursively within a disk, and these disks are arranged
radially around the root, with the edges at each node spaced equally
around the node so as to achieve the best possible angular
resolution. If we are allowed to permute the children of
each node, then we show that a drawing of this type can be made in which
all edges are straight line segments and the area of the drawing is a
polynomial multiple of the shortest edge length. However, if the child
ordering is prescribed, exponential area might be necessary. We show
that, if we relax the requirement of straight line edges and draw the
edges as circular arcs (a style we call Lombardi drawing) then even with a prescribed
child ordering we can achieve polynomial area.

We extend Lombardi drawing (in which each edge is a circular arc and
the edges incident to a vertex must be equally spaced around it) to
drawings in which edges are composed of multiple arcs, and we
investigate the graphs that can be drawn in this more relaxed framework.